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Fastest, average and quantile schedule. (English) Zbl 1435.90066

Italiano, Giuseppe F. (ed.) et al., SOFSEM 2015: theory and practice of computer science. 41st international conference on current trends in theory and practice of computer science, Pec pod Sněžkou, Czech Republic, January 24–29, 2015. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 8939, 201-216 (2015).
Summary: We consider problems concerning the scheduling of a set of trains on a single track. For every pair of trains there is a minimum headway, which every train must wait before it enters the track after another train. The speed of each train is also given. Hence for every schedule – a sequence of trains – we may compute the time that is at least needed for all trains to travel along the track in the given order. We give the solution to three problems: the fastest schedule, the average schedule, and the problem of quantile schedules. The last problem is a question about the smallest upper bound on the time of a given fraction of all possible schedules. We show how these problems are related to the travelling salesman problem. We prove NP-completeness of the fastest schedule problem, NP-hardness of quantile of schedules problem, and polynomiality of the average schedule problem. We also describe some algorithms for all three problems. In the solution of the quantile problem we give an algorithm, based on a reverse search method, generating with polynomial delay all Eulerian multigraphs with the given degree sequence and a bound on the number of such multigraphs. A better bound is left as an open question.
For the entire collection see [Zbl 1303.68019].

MSC:

90B35 Deterministic scheduling theory in operations research
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
90C59 Approximation methods and heuristics in mathematical programming
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